Now we further normalize each row of \(\Gamma \)  such that \(\Vert \Gamma _{i\cdot }\Vert _2= 1\)  \({\rm var}(X_i) = {\rm var}(\Gamma _i{\mathbf {Z}}) = 1\)  for \(1\le i\le p\) . The rest of the proof is essentially the same as Lemma 2 in Cai and Liu [1]} thus we will not go into details. Let
\(\tilde{\theta }_{ij}^\ast = \frac{1}{n_{ij}^\ast } \sum _{k = 1}^n\left(X_{ik}X_{jk} - \tilde{\sigma }_{ij}^\ast \right)^2S_{ik}S_{jk}, \quad \tilde{\sigma }^\ast _{ij} = \frac{1}{n_{ij}^\ast }\sum _{k = 1}^n X_{ik}X_{jk}S_{ik}S_{jk}.\) 
